Two days ago, I read a proof about a basic property of open sets in
Royden’s *Real Analysis*. It says that each open set in $\Bbb R$ is a
countable union of disjoint open intervals. Reading the book, I got
stuck at the countability. I tried to think of a map between each
open interval and a rational number, but I *didn’t* know *which one to
choose*. It took me half an hour to understand from the words that
the *one single* rational number inside the open interval can be
arbitrarily chosen.

Yesterday, I was puzzled at $\mathcal{T} \subseteq \mathcal{B}$ for a
while. On the LHS, I came up with “arbitrary union”; on the RHS, I
thought of “countable union”. It seemed that “countable union” is
more restrictive than “arbitrarily union”. My teacher said that this
is *by definition*, and I realized that I forgot the definition of
Borel sets. He thought that I *didn’t* know $\mathcal{T}_\sigma =
\mathcal{T}$. Finally, I get it.